Restless bandits with controlled restarts: Indexability and computation of Whittle index

Abstract

Motivated by applications in machine repair, queueing, surveillance, and clinic care, we consider a scheduling problem where a decision maker can reset m out of n Markov processes at each time. Processes that are reset, restart according to a known probability distribution and processes that are not reset, evolve in a Markovian manner. Due to the high complexity of finding an optimal policy, such scheduling problems are often modeled as restless bandits. We show that the model satisfies a technical condition known as indexability. For indexable restless bandits, the Whittle index policy, which computes a function known as Whittle index for each process and resets the m processes with the lowest index, is known to be a good heuristic. The Whittle index is computed by solving an auxiliary Markov decision problem for each arm. When the optimal policy for this auxiliary problem is threshold based, we use ideas from renewal theory to derive closed form expression for the Whittle index. We present detailed numerical experiments which suggest that Whittle index policy performs close to the optimal policy and performs significantly better than myopic policy, which is a commonly used heuristic.